Skip to main content

Full text of "Treatise On Analysis Vol-Ii"

See other formats


442       XV   NORMED ALGEBRAS AND SPECTRAL THEORY

is such that D = P~D = DP+, and is therefore of rank m (note that, if y e E and
Xl = (Al + i/)-1 - y, then Xj. e dom(jET*) and y = (H* + il) • Xj.)

14.    A closed unbounded hermitian operator H on E is said to be major ized (resp.
minorized) if there exists a real number c such that (H • x \ x) <: c(x \ x) (resp.
(H - x | x) ^ c(# | #)) for all # e dom(ff). The operator H is said to be positive if
(H • x | x) ^ 0 for all x e dom(jy). Thus jfiT is majorized (resp. minorized) if and only
if there exists c e R such that cl — H (resp. H — cl) is positive.

(a)    Let .ftT be a positive closed hermitian operator on E. The hermitian form
f(xt y) — (x + H-x\y) on dom(H) makes it a prehilbert space, which may be con-
sidered as a dense subspace of a Hilbert space G. Show that the canonical injection
j of domC/HO into E extends by continuity to an injection of G into E, so that G may
be identified with a subspace of E. We denote the scalar product on G again by/(x, y).
Show that there exists a continuous linear mapping B of E into G (where G is con-
sidered as a Hilbert space) such that (x \y) =/(.#• Jt, y) for all xeE and y e G.
Considered as a mapping of E into E, the operator 3 is positive, continuous, self-
adjoint and of norm <|1; we have (B-x\x)^ \\B-x\\2, and B(I + H ) - y = y for all
y e dom(AT). Deduce that A — B""1 — /is an unbounded positive self-adjoint operator,
such that dom(/4) — 5(E), which extends If. Hence the defects of £Tare equal.

(b)    Suppose that the defects of H are equal and finite, say m. If A i is any self-adjoint
extension of H, show that the intersection of Sp(^i) and the interval ]— oo, 0[ consists
of eigenvalues of Ai9 and that the total number of such eigenvalues, each counted
according to its multiplicity, is ^m. (For each compact interval J c ]— oo, 0[, show
that the image of E under the projector (pj(Ai) (Section 15.12, Problem 7), which is
contained in dom(Ai), cannot have dimension >m, otherwise it would contain a
nonzero vector x<=dom(H) (Problem 13(a)); show that this conclusion would con-
tradict the positivity of H.)

15.    Let H be a closed hermitian operator on E, with defect (m, n\ so that —H has defect
(n, m). Show that H is the restriction to E of an unbounded self-adjoint operator A
on the Hilbert sum E© E.

16.    (a)   Let G be an analytic function on the disk B : \z\ < 1, such that ^G(z) ^ 0 in
this disk. Show that the function

K(ii, v) = (G(«) + GW)/(1 - irf)

on B x B is of positive type (Section 6.3, Problem 4). (Use Section 14.11, Problem 18.)
(b) Let F be an analytic function on the half-plane D : Jz > 0, such that ./F(z) ^> 0
for all z e D. Show that the function

is of positive type on D x D (map D conformally on the disk B, and use (a)).
(c) Let S be a denumerable infinite subset of the half-plane Jz > 0. Let C(<p) denote
the angular sector in this half-plane, defined by [0tz\ ^(^z) cos 9?, for 0 < 9? < JTT.
Suppose that there exists a sequence (<jn) of points of S contained in a sector C(<p0)
and tending to 0. Let /be a mapping of S into the half-plane «/z > 0. In order that
there should exist an analytic function F, defined on the half-plane Jz > 0, such that
z) > 0 on this half-plane and such that |F(z)/z| is bounded in every sector C(<p),t follows from Problems 12(e) and ll(c) that A is an eigenvalue